The probability of unusually large components for critical percolation on random $d$-regular graphs

TL;DR

This paper analyzes the probability of large components in critical percolation on random regular graphs, providing precise asymptotics for the size of the largest component near the critical threshold.

Contribution

It derives exact asymptotic formulas for the probability of unusually large components in critical percolation on random regular graphs, improving previous results by Nachmias and Peres.

Findings

Probability of large components scales as A^{-3/2} with exponential correction.

Asymptotic probability for a vertex to be in a large component is established.

Results refine understanding of component sizes at criticality in random regular graphs.

Abstract

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple) graph on $[n]$ and then performing independent $p$-bond percolation on it, i.e. we independently retain each edge with probability $p$ and delete it with probability $1-p$. Let $|\mathcal{C}_{\text{max}}|$ be the size of the largest component in $\mathbb{G}(n, d, p)$. We show that, when $p$ is of the form $p=(d-1)^{-1}(1+\lambda n^{-1/3})$ for $\lambda\in \mathbb{R}$, and $A$ is large, \begin{align*} \mathbb{P}(|\mathcal{C}_{\text{max}}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{\lambda A^2(d-2)^2}{2d(d-1)}-\frac{\lambda^2 A(d-1)}{2(d-2)}}. \end{align*} This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than $An^{2/3}$.